 Solution

Write $p=1/M$ and $v_f=\int {\varphi }^{2}f-{\theta }^2$. The Bernoulli central limit theorem givesIf $M=1$, all proposals are accepted and this limit is degenerate.

Conditional on $N=m$, the accepted sample is iid from $f$. Since $N/n\to p>0$, its size tends to infinity, and the ordinary target-sample CLT therefore givesAssign any fixed value to the estimator on $N=0$; the probability of that event tends to zero, so it does not change this limit. This is the random-count central limit theorem for accepted rejection samples.

For comparison at the same budget of $n$ proposals, Slutsky's theorem rescales this asThe distinction between accepted-sample size and proposal count is essential for the final efficiency comparison.